Kleismic: Difference between revisions

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== Tunings ==

[[File:Kleismic.png|thumb|alt=Kleismic.png|A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.]]

=== Optimized tunings ===

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | ~~Norm~~-based tunings

|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings

|-

! rowspan="2" | ~~Weight-skew\Order~~ !! colspan="2" | Euclidean

! rowspan="2" | !! colspan="2" | Euclidean

|-

! Constrained !! Destretched

|-

! Tenney

| ~~(2.3.5)~~ CTE: ~6/5 = 317.~~0308¢~~ || ~~(2.3.5)~~ POTE: ~6/5 = 317.~~007¢~~

| CTE: ~6/5 = 317.0308{{c}} || POTE: ~6/5 = 317.007{{c}}

|-

! Equilateral

| ~~(2.3.5)~~ CEE: ~6/5 = 317.~~1033¢~~

| CEE: ~6/5 = 317.1033{{c}} (11/61-kleisma)

(11/61-kleisma)

|}

{| class="wikitable mw-collapsible mw-collapsed"

|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.13-subgroup norm-based tunings

|-

! rowspan="2" | !! colspan="2" | Euclidean

|-

! Constrained !! Destretched

|-

! Tenney

| ~~(2.3.5.13)~~ CTE: ~6/5 = 317.~~1110¢~~ || ~~(2.3.5.13)~~ POTE: ~6/5 = 317.~~0756¢~~

| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}

|}

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| 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''3 − ''g'' − 4 = 0 || Close to 13/51-marveltwin comma

|}

=== Other tunings ===

* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.0000{{c}}, ~6/5 = 317.1983{{c}}

=== Tuning spectrum ===

{| class="wikitable center-all left-4"

! ~~EDO~~ generator

! Edo generator

! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]*

! [[Eigenmonzo|Eigenmonzo (unchanged interval)]]*

! Generator (¢)

! Comments

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| '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''

|}

<nowiki />* Besides the octave

<nowiki/>* Besides the octave

~~=== Other tunings ===~~

* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983

== Scales ==

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* [[Cata15]] ([[4L 11s]])

* [[Cata19]] ([[15L 4s]])

~~== Images ==~~

~~[[File:Kleismic.png|alt=Kleismic.png|600x560px]]~~

A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.

== Music ==

@@ Line 113: / Line 113: @@
 == Tunings ==
+[[File:Kleismic.png|thumb|alt=Kleismic.png|A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all edo tunings are shown with vertical lines whose length indicates the edo's tolerance, i.e. half of its step size in either direction of just, and some small edos supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.]]
 === Optimized tunings ===
 {| class="wikitable mw-collapsible mw-collapsed"
-|+ style="font-size: 105%; white-space: nowrap;" | Norm-based tunings
+|+ style="font-size: 105%; white-space: nowrap;" | 5-limit norm-based tunings
 |-
-! rowspan="2" | Weight-skew\Order !! colspan="2" | Euclidean
+! rowspan="2" |  !! colspan="2" | Euclidean
 |-
 ! Constrained !! Destretched
 |-
 ! Tenney
-| (2.3.5) CTE: ~6/5 = 317.0308¢ || (2.3.5) POTE: ~6/5 = 317.007¢
+| CTE: ~6/5 = 317.0308{{c}} || POTE: ~6/5 = 317.007{{c}}
 |-
 ! Equilateral
-| (2.3.5) CEE: ~6/5 = 317.1033¢
+| CEE: ~6/5 = 317.1033{{c}}<br>(11/61-kleisma)
-(11/61-kleisma)
+|}
+{| class="wikitable mw-collapsible mw-collapsed"
+|+ style="font-size: 105%; white-space: nowrap;" | 2.3.5.13-subgroup norm-based tunings
+|-
+! rowspan="2" |  !! colspan="2" | Euclidean
+|-
+! Constrained !! Destretched
 |-
 ! Tenney
-| (2.3.5.13) CTE: ~6/5 = 317.1110¢ || (2.3.5.13) POTE: ~6/5 = 317.0756¢
+| CTE: ~6/5 = 317.1110{{c}} || POTE: ~6/5 = 317.0756{{c}}
 |}
@@ Line 147: / Line 156: @@
 | 13:15:18 (+2 +3) || ~6/5 = 317.0010 || 3''g''<sup>3</sup> &minus; ''g'' &minus; 4 = 0 || Close to 13/51-marveltwin comma
 |}
+=== Other tunings ===
+* [[DKW theory|DKW]] (2.3.5): ~2 = 1200.0000{{c}}, ~6/5 = 317.1983{{c}}
 === Tuning spectrum ===
 {| class="wikitable center-all left-4"
-! EDO<br />generator
+! Edo<br>generator
-! [[Eigenmonzo|Eigenmonzo<br />(unchanged interval)]]*
+! [[Eigenmonzo|Eigenmonzo<br>(unchanged interval)]]*
 ! Generator (¢)
 ! Comments
@@ Line 320: / Line 332: @@
 | '''Upper bound of 2.3.5.13-subgroup 15-odd-limit diamond monotone'''
 |}
-<nowiki />* Besides the octave
+<nowiki/>* Besides the octave
-=== Other tunings ===
-* [[DKW theory|DKW]] (2.3.5): ~2 = 1\1, ~6/5 = 317.1983
 == Scales ==
@@ Line 330: / Line 339: @@
 * [[Cata15]] ([[4L 11s]])
 * [[Cata19]] ([[15L 4s]])
-== Images ==
-[[File:Kleismic.png|alt=Kleismic.png|600x560px]]
-A chart of the tuning spectrum of hanson and cata, showing the offsets of odd harmonics 3, 5, 9, 13, and 15, as a function of the generator; all EDO tunings are shown with vertical lines whose length indicates the EDO's tolerance, i.e. half of its step size in either direction of just, and some small EDOs supporting the temperament are labeled. Comma fractions with corresponding eigenmonzos also labeled.
 == Music ==